# Shhhh - my math secrets

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**STAGE 5:**

Number Sense:

*•1. **Variables & Constants* - the weather is always changing, thus it is a variable, but when it is 90 degrees it is for that moment a constant. *2. **Polynomials & Orders of Operations.* A polynomial is only another way to say a term (or group of terms) that are only added, subtracted or multiplied. But in which order do you attempt the equation? (an equation is a polynomial without an equal sign on the end, by the way) So, 4x + 3xy - 2z is a polynomial. Honestly I always use grocery shopping for these equations. I'm a big couponer, and often you have to buy a certain number of an item for the coupon to work (3/$1off) so I have to multiply the price of an item by 3 before subtracting the $1 to find out the right price! (which finding the answer makes it an Equation, see next item) *•3. **Quadratic Equation -* this is a polynomial WITH an equal sign. So 4x + 3xy - 2z = 0 is a quadratic equation...and, as a BTW, the numbers here are called coefficients because we know their value, whereas the letters (or unknowns) are called variables because in every equation they can mean something different. *•4. **Parenthesis.* Parenthesis, in a quadratic equation, interrupt the order of operation. Even if addition/subtraction is inside the parenthesis you do them first. So, with 4(2+7) = x you would first add inside the parenthesis to get 4(9) = x so 36=x. Simple. *•5. **Ever Expanding Binomials.* A binomial is a quadratic equation is an expression containing 2 terms. So, (x+y) is a binomial. Sometimes we need to expand the expression in order to figure it out.....basically this is a combination of the parenthesis rule And the Factorization rule...so (*a* + *b*)^{3} = *a*^{3}+ 3*a*^{2}*b* + 3*ab*^{2} *+b*^{3} , you look to the power, then you separate the term (a+b) the first thing you do is add the power individually to each part of the term so a to the power + b to the power, (a+b)(a+b) how ever many times equal the power so this is really just another way to do foil. Have to remember the middle parts, which would be (number=to power)x (a raised to the power-1)b + (number=to power)x a (b raised to the power-1) that's all. It looks way complicated but really the rule is simple. Add in the constants. *•6. **FOIL* - this is an easy peesy way of treating a quadratic equation with 2 parenthesis multiplied together....This is the short cut of the previous step. If they get a little overwhelmed, teach foil. Some math teachers don't teach foil and almost hate foil. But I think you could almost skip step 9 as long as they have a solid understanding of FOIL then they will be able to get to the answer in the correct way, and that's all that is needed! FOIL stands for First, Outer, Inner, Last. So if you have (x+y)(z+b), then you would multiply the first positions (xz), then the outer (yb), then the inner (yz) then the last (yb). So the long hand would be

Xz+yb+yz+yb why would you want to make it so messy? Well usually it's so you can add some of the terms. So if the equation was (x+2)(x-3)=24, you would foil to get (x^2)-x-6=24 add the 24 to both sides to get (x^2)-x-30=0

*•7. **UnFOIL* - ok hang in there with me....I know it seems as if there isn't a real world application for the last few steps but there really is, once you understand the process you can start applying it. Especially when dealing with AREA like Area of a Rectangle. First, we've learned to breakdown the quadratic equation in order to bring the equation to zero (by adding 24 to both sides in the last step) and simplifying. But now that you are left with a messy equation you can re-simplify it essentially doing the reverse of FOIL. Here is where the practical application comes in....(example) The length of a rectangle is 6 inches more than it's width. The area of the rectangle is 91 square inches. Find the dimensions of the rectangle. First draw the rectangle. The length (long sides) are x+6, the width is x, If you recall area of a rectangle equation from Stage 1, it's length x width. So we multiply x(x+6) = 91. Multiply the x through the parenthesis, so you get x^2+6x = 91 we want the equation to equal zero, so subtract 91 from both sides; x^2�91=0. With reverse foil, or the expanding quadratic step, we split the equation up as follows, the x^2, so we know (x )(x )=0. To get the negative number out of the way we know one term will be + and the other - so, (x- )(x+- )=0, so now we are left with the 6 and the 91, we now need to find 2 numbers that when multiplied will equal 6 and the two together multiply to 91. So we know the negative position can by a 7 (as x-7x will be 6) and the second will be 13 as 7x13=91! So the result is (x-7)(x+13) and you can double check it with FOIL to make sure it results in the original messy equation. *•8. **Double Checking Answers.* It can be easy, when working with polynomials, and quadratic equations, to get a little lost in the shuffle. There are many ways of checking your answer.

**Parts of A Whole:**

•1. Divide Fractions

•2. Fractions = Decimals

•3. Identifying place value of decimals

•4. Parts of 100 = % = x/100

•5. Make a number line and identify decimals, fractions and % on it.

•6. *Scientific Notation=* this is a way to write a number that is actually a tiny decimal. When there are a lot of zero's before the number (like, 0.00007063000) it can be tedious to write this number through all of your work. So instead you would write the number x 10^how many zero's. So, using the example. I would convert it to 7.063 x 10^-5 -because I moved the decimal over 5 spaces (because we're working on a tiny decimal we use the negative)! That's all, super easy. So 300 would be 3x10^2, 4,000 = 4x10^3, 5,720,000,000 = 5.72 x10^9, etc.

**Statistics & Probability:**

*•1. **4 Quadrants / Ordered Pairs.* This is a build off of using quadratic equations, but you can choose to learn this first. Basically you have (x, y) and from this you can graph on the x,y planes. But this is just ¼ of a cross, if you have a full cross with grids on all sides then you have 4 quadrants. Essentially you have 2 number lines that intersect, one going up/down, the other going left/right. When using the number lines you can see that the upper left quadrant is (-x,y) upper right is (x,y) lower left is (-x,-y), and the lower right is (x, -y).

**Geometry:**

•1. Making a Cube and Rectangular Box - changing 2D to 3D images. Experiment with the different units of measurement between 2D and 3D.

•2. Understanding Volume and it's measurements.

•3. Discovering Pi, play with Diameter, Radius and Circumference of a Circle.

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**Add your quick reply below:**

- KickButtMama

Shannon on Nov. 21, 2011 at 3:08 PMOk, as I've said previously, these steps are not set in stone - you can jump around. I just think this is the easiest way to learn the information without muddying the waters.

STAGE 4:Number Sense:•1.Multiplication & Division- getting comfortable with manipulating integers by multiplication/division and checking the results with addition/subtraction.•2. Add/Subtract Positive/Negative Integers

•3. Multiply/Divide Positive/Negative Integers

•4.Exponents- Raising a number to a power - repeated multiplication.•5.Prime/Composite Numbers- a number that can only be divided evenly by 1 or itself. So, 2, 3, 5, 7, 11, 13 and 17 are all prime numbers. Any number that can be divided evenly is called a composite number. This is so important it has a name: Fundamental Theorem of Arithmetic = The basic idea that any integer above 1 is Either a Prime number or can be made by multiplying prime numbers together.•6.Prime Factorization- This is finding out what prime numbers multiply to give you a different or ‘given' number. So if I gave you a 6 and told you to find the Prime Factors, then they would be 2 and 3. 2 and 3 are prime numbers and if you multiply them together you get the ‘given' number of 6. So what are the prime factors of 12? Well....divide it by 2 (prime) and you get 6 (not prime) so 2 and 6 are not prime, but you can divide 6 by 2 and get 3 (which is prime) so you would say 2x2x3 = 12 or 2^2x3=12. Therefore the prime factors of 12 are 2, 2, and 3. If the student is good at wrote memorization then you can have them memorize some of the basic prime factors, but for everyone else it's just a bit of playing w/ the number. Is the number even or odd? If odd then 2 won't work so you'll start by dividing it by 3. Etc. You can also break a number down to make it easier to manipulate: 90 can be broken into 9x10. Prime factors of 9 are 3 and 3, prime factors of 10 are 2 and 5; so the prime factors of 90 are 3, 3, 2, 5. This is really helpful when doing LCF - knowing how to find the lowest factor between 2 numbers. Uses of Prime numbers? Well believe it or not, one of the best uses today for prime numbers is Cryptography - (secret codes). PF can be done using lego's or other manipulatives - Get enough pieces to make the number (ie. 12) if you can only group the pieces by 1's or the original number then it is prime, but if you can make ANY other groups evenly (w/o pieces left over) then these groups are the factors. 12 can be broken into groups of 2, 3, 4, 6, 12, or 1 so it is not prime but each of these quantities are it's factors!Parts of a whole:•1. Fractions - my favorite is pizza (in more ways than 1) but it's easy to demonstrate fractions with a pizza.

2. Addition & subtraction of similar (same denominator)

•3. Simplify Fractions

•4. Multiply like fractions

•5. Lowest common Factor

Statistics/Probability:•1. Using various graphs to answer questions.

•2. Median, Mean & Mode.

•3. Displaying data in various graphs - histogram, circle graphs, etc.

•4. Intro to Excel - using a spread sheet of terms to make a graph

Geometry:•1. Lines, Rays, Segments

•2. Parallel & Perpendicular lines

•3. Angles - using various tools to make specific angles and lines (ruler, protractor, etc)

•4. Triangle & Quadrilateral - all angles sum up to 360 degrees

•5. Area & Perimeter of Squares & Rectangles = Comparing the Area formulae for Triangle, parallelogram and rectangle. 2 of the same triangles make a parallelogram w/ twice the area; a parallelogram is compared w/ a rectangle of the same area by cutting and pasting a right triangle on a parallelogram.