NUMBER SENSE (MARKED 2009/06/07)




BASICS
Adding and Subtracting are the two most basic forms of mathamatics.
When you are adding, you are generally putting two numbers together. Therefore, you are making a number larger. When you are subtracting, you are generally taking a number away from a certain number. Therefore, you are reducing a number and getting a smaller product. However there are some exceptions.(Refer to rules) When adding or subtracting numbers, there is the event where you will be dealing with positive and negative numbers. In this case, a number could increase as you subtract or decrease as you add.
Examples
2 + 4 =
6the number got larger.
4 - 2 = 2 the number got smaller.


The two other basic forms of mathematics are multiplying and dividing.
Multiplying is when you basically have a certain amount of groups of a certain number.(i.e. 3x8 = 3 groups of 8) Multiplying is an easier way of doing something like 2 + 2 + 2 + 2 = 8. Dividing is basically determining the amount of times a number can fit into another certain number. (i.e 12/6 = 2 can fit into 12 two times) Another way to divide is by using a variable. Use the question 45/5 as an example. Therefore you would think of it as ? x 5 = 45. So, you would figure out what number times 5 equals 45. Since 9x5 = 45, 45/9 = 5.



RULES
When adding and subtracting, these rules apply only for when the signs are directly beside each other.

+ and + = a positive
- and - = a positive
+ and - = a negative
- and + = a negative


When multiplying and dividing, these rules apply for whatever the numbers are when bothe multiplying and dividing. (They don't have to be beside each other)

+ and + = a positive
- and - = a positive
+ and - = a negative
- and + = a negative




What is Number Sense?
We define number sense in the school environment as, an understanding that allows students to approach concepts, ideas, and problems concerning numbers differently, according to their backgrounds, experiences, studies, etc. These approaches students use are often referred to as heuristics.



Exponent Laws

There are 3 exponent laws:

1.Multiplication Law
2.Division Law
3.Power of a Power Law

1.Multiplication Law: When multiplying powers with the same base, you add the exponents together.
Example: 3^4 x 3^5 = 3^9 (the base stays the same)

2.Division Law: When dividing powers with the same base, you subtract the exponents.
Example: 6^14 / 6^3 = 6^11 (the base stays the same)

3.Power of a Power Law: When raising a power to a power you multiply the exponents.
Example: (8^2)^6 = 8^12 (the base stays the same)



BEDMAS

Bedmas (an acronym) is form of order of oprations, these steps must be used in most integer equations.


  • Brackets
  • Exponents
  • Division
  • Muliplication
  • Addition
  • Subtraction


Following the order means you have to calculate whatever is inside the brackets first (eg. 45 x 15
=?). Then you must calculate the exponent into the equation. Then divide the numbers if possible. Then multiply if possible. then add if possible. Then when there is only addition and subtraction left to perform, we work from left to right.

Examples
(Trevor's Example)
12+(-5)+(-7)
=12-5-7
=7-7
=0
As you can see after each step it is imporant to put an equals sign.
When dividing fractions the rule with the signs is the same when adding.
(-8)/(-8)=1 because the 2 negatives equals a positive.



(Amanda's Example)
If I ask you to find 2 + 3 external image multiply.gif 6 + 4 then you have to make a decision.

Do you read from left to right, first adding 2 and 3, and get
5 external image multiply.gif 6 + 4 = 30 + 4 = 34?
Do you read from right to left, first adding 6 and 4, to get
2 + 3 external image multiply.gif 10 = 2 + 30 = 32?
Do you perform the additions first and then the multiplication
5 external image multiply.gif 10 = 50?
Do you perform the multiplication first and then the additions
2 + 18 + 4 = 24?
Any of these might be considered to be "the right answer".
Faced with this ambiguity, we need to agree on a procedure so that we all calculate 2 + 3 external image multiply.gif 6 + 4 the same way. The agreeded on "rule" is that you multiply and divide before you add and subtract. Sometimes problems are even more complicated than that, like the expression you have in part 2. If you remove the brackets "[,]" and parentheses "(,)" you see that there are many ways you might calculate the resulting expression. The brackets and parentheses are used to group parts of the expression together.
The procedure is to look at the expression in the most "inside" brackets or parentheses. (This is the B in BEDMAS, B for bracket.) In your example this is
6+1-3x2
perform the multiplication first and then the addition and subtraction.
6 + 1 - 3 x 2 = 6 + 1 - 6 = 1
Hence the problem becomes
5 + 5 - 4 + [6 x 3 - 1 - 5 + 9]
Now the expression most "inside" brackets is
6 x 3 - 1 - 5 + 9 = 18 - 1 - 5 + 9 = 21
Finally this gives
5 + 5 - 4 + 21 = 27
The term BEDMAS is just a memory devise to help us remember the order.



EXAMPLE:


(3 + 6) - 8 × 3 / 24 + 5

Following BEDMAS, we need to start with B-brackets,

(3 + 6) - 8 × 3 / 24 + 5
= 9 - 8 × 3 / 24 + 5

Then E-exponents, none, followed by DM-divide multiply (left to right),

= 9 - 8 × 3 / 24 + 5 (multiply)
= 9 - 24/24 + 5 (divide)
= 9 - 1 +5

And finish with AS-add subtract,

=9 - 1 +5
=8 +5
= 13







SCIENTIFIC NOTATION


Scientific Notation is a way of writing numbers that accomodates for values that are too large or too small to conveniently be written in simple decimal notaion.

a x 10^b

In scientific notation, a number has the form a x 10^b, where "a" is greater or eqal to 1 but less than 0, and 10^b is a power of 10.
These numbers can be multiplied and divided from following the laws of exponents.


How does Number Sense begin? (Early Childhood)
Kindergarten: At an early age, close to 3 years old, children start to learn and develop small counting skills. They can eventually easily recognize groups of 1, 2, and 3. but when numbers get larger, different mental strategies must be used. Children learn to see higher number groups as not just on group, but many evenly numbered groups put together. For example: Seeing a group of 10 as two groups of 5. This gives children an understanding of (in this case 10) is made of multiple parts. Soon these small groups are instantly recognized and instant addition of these two groups forms.

First Grade: Counting words and strategies are linked to symbols. Kids learn to clearly grasp the concept of operation symbols, such as subtraction and addition indicators. They learn how to differentiate these symbols, how and when to write them, and how a number increases or decreases depending on what symbol is being used.




Place Value
Numbers, such as 84, have two digits. Each digit is a different place value. The left digit is the tens' place. It tells you that there are 8 tens. The last or right digit is the ones' place which is 4 in this example. Therefore, there are 8 sets of 10, plus 4 ones in the number 84. Place value charts range from ones, to billions. The individual digits in a very large number all lie under a certain spot in the place value chart. Using place value, large numbers can be understand at a young age.

Standard form:
Numbers can be represented in many ways, but standard form is usually the shortest and easiest. Here are some numbers expressed in different forms with their standard forms along side them.

one billion, sixty million, five-hundred twenty-thousand= 1 060 520 000
four-hundred sixteen-thousand,seven-hundred thirty-one= 416 731
60 000+10 000+450+1= 70 451

In each of these cases, what value does the value does the digit "5" have?
51, 567= fifty-thousand (50 000)
1 080 775= five (5)
777 509 701= 500 000 (five hundred-thousand)


Decimal Numbers
Zero and counting numbers (1,2,3,...) make up the set of whole numbers. But not every number is a whole number. The decimal system lets us write numbers of all types and sizes, using a symbol called the decimal point.
As you move right from the decimal point, each place value is divided by 10.
The decimal system lets us define the value of a number to a greater extent. In our number system, digits can be placed to the left and right of a decimal point, to indicate numbers greater than one or less than one. The decimal point helps us to keep track of where the "ones" place is. It's placed just to the right of the ones place. As we move right from the decimal point, each number place is divided by 10.
We can read the decimal number 127.578 as "one hundred twenty seven and five hundred seventy-eight thousandths". but we should usually read is like: "one hundred twenty seven point five seven eight."



Examples
This is how to write numbers in decimal form:

(6 x 10) + (3 x 1) + (1 x 1/10) + (5 x 1/100)

63.15
Five hundred forty-eight thousandths
0.548
Five hundred and forty-eight thousandths
500.048

Hint #1: Remember to read the decimal point as "and"--.
Hint #2: When writing a decimal number that is less than 1, a zero is normally used in the ones place:

*0.526 not .526*










ESTIMATING/ROUNDING DECIMALS
Estimating is an important part of mathematics and is very handy. It is used in estimating amounts of money, lengths of time, distances, and many other physical quantities.
Rounding is a kind of estimating.
To round off decimals and whole numbers:

1. Remember that "5" is the middle number.
2. Look to the number to the right of the digit of the certain place value you are rounding to.
3. If that number is equal or more or than "5" it will be rounded up, but if it is lower than "5" the number will be rounded down.






Examples:



To round 14,769.3352 to the nearest hundred
Find the rounding digit, "7". Look at the digit one place to right, "6". Six is more than 5, so this number needs to be rounded up. Add one the rounding digit and change all the rest of the digits to the right of it to zero. You can remove the decimal part of the number too. The result is 14,800.


Round 7 890 to hundreds place:
Answer: 7 900, the number ro the right of 8 is "9". "9" is larger than 5 so it will be rounded up.

Round 1 567 897 to hundred thousands place:
Answer: 1 600 000, the number to the right of "5" is 6. 6 is larger than 5 so it will be rounded up.




VARIABLES

Variables have three factors. First, a variable is a letter or a symbol used to represent an unspecified number. Secondly, a variable takes the place of an unknown value. Variables are most commonly represented with "x"

EXAMPLES


(-2) + 7
= 5
b) 20x
= (20) (-2)
=- 40



Expressions With Two Variables

a) xy
= (-8) (2)
= - 16
b) x+6-y
= (-8) +6-2
=0




Number Sets

Real Numbers: Any number that you can think of is a real number.
Integers: Positive or Negetive whole numbers
Whole Numbers: Positive whole numbers including zero
Natural Numbers: Positive whole number (counting numbers)

Rational Numbers

Numbers written in the form a/b where b cannot be zero
Including all fractions, all integers, all terminating decimals and all repeating decimals.

Irrational Numbers

Simply means 'not rational'
Numbers that cannot be expressed as fractions and have neither terminating or repeating decimals




Percents

A percent is a fraction that has a denominator of 100
Percents are all in relation to decimals and frctions
We can change: a fraction to a percent, a decimal to a fraction, a percent to a decimal, a decimal to a percent and a percent to a fraction.

Changing a fraction to a percent: 17/100 = 17% or 25.7/100 = 25.7%
Changing a decimal to a fraction: 0.12 = 12/100 = 3/25 ( always put in lowest terms)
Changing a percent to a decimal: 15% = 0.15 (simply move the decimal place two places to the left)
Changing a decimal to a percent: 0.5= 50 (to convert from decimal to percentage, just multiply the decimal by 100)
Changing a percent to a fraction: 50% = 50/100 (since the word percent means per 100 you put the number of the percent over 100)

Ratios and Rates

What is a Ratio?

  • A ratio is a comparison between two numbers
  • You can write ratios as a fraction (5/8) or with a colon (5:8)

Examples

Alex has a bag with 8 CD's, 3 marbles, 2 books and 3 apples.

1.a) What is the ratio of the CD's and apples?
Answer= The ratio of the Cd's and the apples is 8:3 or 8/3

b) What is the ratio of the CD's to the rest of the items in the bag?
Answer = 8:8 or 8/8

2. What is the ratio of the Green Stars to the Blue Stars?

**

    • 6 : 2
















Works Cited:

Harvard Newsletter. (2008). Harvard. Retrieved December 17, 2008, from http://edletter.org/insights/stages.shtml
Bobis, J. (1996). Visualization and the development of number sense with kindergarten children. In Mulligan, J. & Mitchelmore, M. (Eds.) Children's Number Learning : A Research
(2005). Math.com. Retrieved December 18, 2008, from http://www.math.com/
Wikipedia.com. (1996). Bedmas. Retrieved December 17, 2008, from http://en.wikipedia.org/wiki/Bedmas