Math notes
Pythagorean Theorem, Squares and Square Roots
Key vocabulary:
- Square number
- Perfect square
- Square root
- Leg (of a right triangle)
- Hypotenuse
- Pythagorean theorem
- Pythagorean triple
What do you need to know:
- The side length and area of a square:
(but in a square they are the same so:
)
This means that the side length of a square is the square root of its area
- Know these perfect squares:
=9 / / /
/ = 5 / =144 /
/ = 6 / /
49 / = 7 / 196 / = 14
/ / / = 6
- The Approximate square root of a number:
How do we estimate roots that are not perfect squares?what about: The roots must be between 3 and 4! We can also use a numberline in order to estimate the root so must be between 1 and 2 but closer to one so I put it on the numberline:
Using the PythagoreanTheorem
If we draw a square on each of the legs of the right triangle their total area is equal to the area of a triangle written on the hypotenuse
We use this formula:
We can use this formula to solve triangle problems
Example:
Find the missing dimension:
Substitute the values you know into the formula:
A Pythagorean triple is a set of three whole numbers that satisfies the Pythagorean theorem
Which is a Pythagorean triple 3,4,5 or 3,5,7
To Solve: substitute them into the formula and if the left and right sides are both equal it IS a Pythagorean triple therefor 3,4 and 5 are a Pythagorean Triple while 3,5, and 7 are not.
= =25 / = =39=25 / =34
Fractions, Percent, Ratio and Rates
How to find an equivalent fraction (or rate or ratio):Step 1. Multiply (or divide) the top (numerator) of the fraction by a number: / 2 × 3 = 6
Step 2. Do the same operation to the bottom (denominator) of the fraction: / 3 × 3= 9
Step 3. Write the answer in the form of a fraction: /
So: / / we multiplied both numerator and denominator by 4
8:16= 4:8 / we divided both terms of the ratio by 2
☺ Number 1 rule ** “What you do to the term one you must do to the term two!!”**
OR “What you do to the numerator you must also do to the denominator!!!”
How to change a fraction to a decimal
Step 1: Divide the numerator by the denominator /
How to change a fraction into a percent
Step 1: find an equivalent fraction over 100 /
Step 2: the numerator of the fraction over 100 is equal to the percent / = 60%
How to change a decimal into a percent
Step 1: multiply by 100 / 0.5 × 100 = 50 %
How to find the percent of a number
Step 1: change the percent into a decimal (divide by 100)
Step 2: multiply the decimal by the number
How to add (or subtract) fractions:
Step one: make sure the fractions have a common denominator /
Step two: add (OR SUBTRACT) the numerations and write the sum as the numerator of a new fractionover the same denominator as the question /
Step three: write the fraction in lowest terms /
How to multiply fractions:
How to divide fractions
DON’T!!! Instead multiply by the reciprocal /
Cross Multiplication
Useful for any equation that can be written in this format:
To solve:
/ Step 1: multiply across the equals sign so
the numerator of fraction A times the denominator of fraction B
is equal to
the denominator of fraction A times the numerator of fraction B.
Step 2: Solve the equation using algebra
Use this for algebra, finding the percent of a number, and finding missing terms in a rate or ratio and many more…
How to find the percent of a number or what number is a percent :
What is 45% of 70?Remember 45% is a part of 100% and 70 is a whole. A fraction is: and we are trying to find a part which we make x. /
What percent is 30 of 65?
Here we are trying to find the percent so we make that x. /
Using formulas for the Surface Area and volume of cylinders and prisms
Shape: / Surface Area: / Volume:cylinder / SA= /
prism / SA=( /
Things to remember:
A cylindar has a circular base:
Circumference of a circle: / /remember / /
and / /
Area of a circle:
The bases of all prismswill be shaped like: /
Area of a rectangle /
Area of a parallelogram /
Area of a triangle: / /
Example
:
Find the surface area and volume of this cylinder: / SA=
written in terms of
V
written in terms of
Operations with integers:
Adding and subtracting: USE A NUMBER LINE
adding a negative number. subtracting a positive number
subtracting a positive number. adding a positivenumber.
move left on the number line go right on the number line
Step 1:start at the first number
Step 2: figure out your direction
Step 3: count spaces of the next number. Where you land is your answer.
example: -20 + -5= start at -20 count 5 spaces to the left because + -5 goes that way.
-21, -22, -23, -24, -25. So -20 + -5= -25
4 - -5 = move right 5,6,7,8,9. so 4 - -5= 9
Multiplying or dividing integers
there is only one rule:
when the signs are the same the answer is positive
and when the signs are different the answer is negative.
examples:
( -3) ×(+4) = -12 (-5) ×(-3) = +15. (-30) ÷(-5) = +6 (+6) ÷(+2)= +3
(+5) ×(-6) = -30 (+6) ×(+3) = +18. (+28) ÷(-7) = -4 (-14) ÷(+2) =-7
Don’t forget to use BEDMAS to solve an equation:
= -3 + -4 ×-5 -(-10 + 5)
= -3 + + 20 -(-5). first I solved the part in brackets and did the multiplication
= -17+ 5. then I did the addition and then the subtraction
= -12
ALGEBRA!!!!!!!
Writing an equation
translate the words into numbers:
four more than +4
three less than -3
two times a number 2x
a number x (n or any other letter)
is =
seven less than six times a number 6x - 7
four more than five times a number is thirty-five 5x + 4= 35
a man divides the money in his wallet between 4 x/4
Important vocabulary:
coefficient the number in a term that is multiplied or divided into the variable
variable the unknown quantity represented by a number
constant term the number term that has no variable
To solve:
6x + 7 = 43
Step 1: start with the constant term
Step 2 line up your equal signsand do the same things to both sides in order to preserve equality
Step 3: use the opposite operation to isolate the variable
Step 4: write the answer in the form x = # (the variable equals some number)
6x + 7 = 43. start with the constant term 7
6x + 7 -7 = 43-7 subtract 7 from both sides
6x = 36. now look at the coefficient 6
6x ÷6 = 36 ÷6 divide both side by 6
x = 6. you are left with the answer!
and another:
y/4-5 = 15. start with the constant term -5
y/4 -5 + 5 = 15 + 5. add+5 to both sides of the equal sign
y/4 = 20. now look at the coefficient 4
y/4×4 = 20 ×4. multiply both sides by 4
y = 80. you are left with 80 THE ANSWER!!!!! easy right ;)
and another:
4x = 84 no constant term so look at the coefficient
4x ÷4 = 84 ÷4. divide both sides by 4
x = 21. you are left with the answer!
and another:
k/3 = 5.
k/3×3 = 5 ×3. multiply both sides by 3
k = 15
Graphing relations
Substitute values for one variable into the relation to generate ordered pairs record them in a
TABLE OF VALUES
4x = y. y=3x-1
for y=3x-1 we can have ordered pairs (-2,-7) and (0,-1)
In the pair the (x,y) the x represents the coordinate on the x-axis and the y value represents the value on the vertical y-axis
we can then graph those ordered pairs on the coordinate pairs:
A(2,5)
B(1,2)
C(0,-1)
D(-1,-4)
a linear relations graph will be a straight line when you connect the dots.