by Courtney Coffman 7 years ago
358
Elementary Mathematics
by Courtney Coffman 7 years ago
358
Quadratic Equations
The way that we multiply binomials is the same way we multiply any numbers, we make an area model.
Steps:
Factoring Steps:
Rational and Irrational Numbers
Rational Numbers: Can be written as a fraction (Decimal can repeat)
Irrational Numbers:
Absolute Value
|-3| = 3
| 5| = 5
| 3- 5| = | -2| = 2
When there is an X inside the absolute value, split it into two for a positive and negative answer
| x - 2 | = 6
x - 2 = 6 x - 2 = - 6
x = 8 x = -4
Fractions
Numerator: How many pieces
Denominator: What size pieces
How to find common denominator: Factor each denominator, multiply each by the factors that are missing
To multiply fractions: cross multiply within the problem to simplify and then multiply straight across
25- 3/8 = 24 8/8 - 3/8 = 24 5/8
13 + 5/11 = 13 5/11
14 3/7 + 30 2/7 = (14 + 30) AND 3/7 + 2/7 = 44 5/7
42 3/5 - 20 2/5 = (42-20) AND 3/5 - 2/5 = 22 1/5
Comparing Fractions: Which is bigger
Modeling:
Operations with modeling:
Percents
10% = Move the decimal over 1
Multiples of 10 = take the value from 10% and multiply it by the number in the tens place of this number
5% and multiples of 5= take half of the 10% value and add it to the total of 10% and the multiple of ten
Harder Numbers:
37% of 473
10% = 47.3
1% = 4.73
37% = 4.73 X 37
Square root
Perfect Square: When a square root comes out to a whole number
For anything that isn't a perfect square, we use perfect squares to estimate them.
Estimate the square root of 53
Order of Operations
G (Groups)
E (Exponents)
M/D (Multiplication/Division Left to Right)
A/S (Add/Subtract Left to Right)
Finding Groups:
Remember: Unless a negative is inside (), the exponent does not apply to it
File Types and Techniques
File Type: What the problem is
Technique: How you handle the exponents (examples correspond with the order above)
Show: Draw out each of the values in the problem so that the largest exponent is one.
Examples of each type of problem:
Multiply same base (add exponents): X^2 * X^3 = X^5
Divide same base (subtract exponents): X^4/X^2 = X^2
Raise a power to a power (multiply exponents): (X^2)^2 = X^4
Properties of Addition and Multiplication:
Commutative: States that the order of terms in a series of repeated addition or multiplication doesn't matter.
3 + 4 + 6 = 6 + 3 + 4 3 x 4 x 6 = 6 x 3 x 4
Associative: You may group any terms within a series of repeated addition or multiplication.
3 + 2 + 5 = (3 +2) + 5 = 3 + (2 + 5)
Distributive: When you have a term being multiplied by a binomial, you can multiply the outside term by each of the inside terms.
3 ( 10 + 4 ) = 3(10) + 3(4)
Identity Property: Identifies the value that keeps the number the same (Plus and Minus = 0, Multiplication and Division = 1)
5 + 0 = 5
5 - 0 = 5
5 x 1 = 5
5/1 = 5
Alternative Algorithms:
Lattice Addition: Line the numbers up vertically. Place a box around where the answer would go and add diagonals to the box. add each digit and place the solution in the box with each place value split by the diagonal. Add down the diagonal to get final solution.
Lattice Multiplication: Place one number of the top of a box and one on the right side vertically. Divide the box into small boxes and add diagonal lines. Multiply each number and place the solution in the boxes. Add along the diagonals for final solution.
Friendly Number Addition: When adding two numbers, take away some amount from the first number to make the second one a multiple of 10 and then add the numbers.
Friendly Number Subtraction: Add the same amount to both terms to make the second number a multiple of 10 and then subtract the number.
Left to Right Addition: Add the digits in a multi-digit problem from left to right including their place value. This is a modified expanded form.
Area Model: Write one number in expanded form on top and the other in expanded form on the right side. Multiply each section of the expanded number to the corresponding sections from the other number and add all solutions to receive final solution.
Scratch: Add each number in a sequence until you get to 10. Place a slash and write how many remain. Write the final remainder at the bottom and carry the number of slashes as a value into the next column. Repeat until a final solution is received.
Expanded form: Can be done for +, -, and x
143 = 100 +40 +3
221 = 200 + 20 +1
solution = 300 +80 + 4 = 384
Integers:
Models:
Model:
Addition
3 + (-4) = +++
- - - - = -1
Subtraction (Underline = take away)
5 - 2 = + + + + + = 3
Multiplication
3(2) = + + + + + + = 6
Diagram:
Addition
-23 + -74
- -- add - 23 +74 = 97 -97
Subtraction
Keep Change Change: Keep the first term, change the sign to its opposite and the final term to its opposite.
Multiplication
Different signs = Difference (Value is negative)
Same signs = (Value is positive)
Greatest Common Factor
Factor: The different #'s that when multiplied give you the number you are factoring
30 : 5 x 6 12: 6 x 2 . GCF: 6
3 x 10 1 x 12
1 x 30 3 x 4
2 x 15
Least Common Multiple
12 : 24, 36, 48, 60
30: 60
LCM: 60
Factors:
Prime- Only two factors (one and itself)
eg... 2, 3, 5, 7
Composite- more than two factors
eg... 4, 6, 8, 12, 24
Prime Factorization:
Tree:
12
^
2 6
^
2 3
12: 2^2 x 3
Upside Down Division:
2| 18
3| 9
3
2 x 3^2
GCF Using Prime Factorization: List all of the common factors using the value with the smallest exponent. Multiply them to receive your solution
LCM Using Prime Factorization: List all of the factors for both numbers. Use the values with the larger exponents. Multiply them to get final solution.
Introduction to Bases:
We most often work out of base 10, which means that one "long" as mentioned before, contains 10 units.
The amount of units it takes to make a long is where we identify what base we are in.
Example: Base Seven would mean that one long contains 7 units, and one flat contains 49 units.
The base you are in is written in word form as a subscript. It is not needed to write the base when you are in base ten.
Diagramming Numbers in Different Bases:
We can write numbers in different bases the same as we would in base 10: the digit in hundreds place represents how many flats we have, the tens place represents how many longs we have, and the digit in the ones place represents units.
For the purpose of notes...
D = flat
| = long
* = unit
Example:
23seven = | | *
245six = D D |||| *****
Place Value:
When looking at a multi-digit number, we look at each digit as a number with a specific place value.
We commonly work in Base 10, which means we have 9 digits before we have to add a second digit and a place holder.
Once we have multiple digits, we can identify the different place values using blocks.
In our blocks, we have small cubes which are equal to one unit each. These units will represent the digit in the ones place of the number.
We also have long rods which would represent 10. These are called "longs" and represent the number in the tens place of a number.
We also have squares, which represent 100. These are called "flats" and they represent the number in the hundreds place of the number.
Finally, we have cubes. These cubes represent 1,000. They represent any digit in the thousands place of a number.
After we reach the point of a cube, it becomes our new unit and the pattern of Flat, Long, Unit continues, and we use commas to reflect that the next "unit" is actually a cube.
Problem Solving:
UnDevCarLo:
Understand the problem
Develop a plan to solve it
Carry out your plan
Look back at your answer
Adding Numbers in Different Bases:
When you add numbers in bases other than base ten, you follow the same protocol as normal addition.
The only difference is if the number that results from the addition has a digit larger than the base.
At that point, you must then exchange it for the next value up, whether it would be a long, flat, or cube.
Ex:
12four + 10four = 22four
| ** + | = ||**
57nine + 28nine = 86nine
|||||******* + ||******** = ||||||||******
Scratch Method:
Line up numbers vertically
Add each digit in the ones place, add until a long is formed, and then place a scratch on it. Continue with all columns. (Italices are equivalent to a number being scratched out for typing purposes)
1 2
25six
12 1
4
+ 23 3
1 1 3 six
Converting to Base 10:
When converting a number in a base other than ten to base ten, we are simply figuring out how many units are in the number.
Show: Show means we must diagram the process we take to get to the answer.
24five to base ten = 14
||**** Each long is equivalent to 5 because that is the base we are in
5 + 5 + 4 = 14
Solve: This method may include the diagram if it is helpful, but requires grouping (multiplication)
25seven to base ten = 19 Each long is equivalent to seven because it is the base
||*****
2(7) + 5(1) = 14 + 5 = 19
Converting From Base 10:
For small numbers we can simply count up all of our units and group them into longs and flats as they appear.
For larger numbers, this can be difficult, so we do upside down division.
We figure out how many longs will make up our number by using division.
37 to base 4: 4|_37 r
4|_9 1
2 1
Answer reads from the bottom number up the remainders in order. 211four
Subtracting in Bases:
Exchange when you don't have enough units
Use the wording of "take away"
For typing purposes- values taken away will be represented as non-bolded
25six - 12six = 13six
|| *****
34six - 15six = 15six
||(******) ****
