Class Notes

*HOW TO FIND THE AREA AND CIRCUMFERENCE OF A CIRCLE

VOCABULARY:

RADIUS – the distance from the center of a circle to any point on the circle itself is exactly the same.

DIAMETER – the length of the line through the center that touches two points on the edge of the circle.

AREA FORMULA: 3.14 x r2                           CIRCUMFERENCE FORMULA: 2 x 3.14 x r

Step 1: find the formula

Step 2: replace the r with the value of the radius. If you are given a diameter than divide the diameter by 2 and quotient is the radius.

Step 3: multiply the numbers together

example: Find the area of a circle with a radius of 3 meters.

step 1: the area formula is 3.14 x r2

step 2: the radius is 3

step 3: multiply 3.14 x 3 x 3

example: Find the circumference of a circle with a radius of 3 meters

step 1: The circumference formula is 2 x 3.14 x r

step 2: the radius is 3

step 3: multiply 2 x 3.14 x 3

example: Find the circumference of a circle with a diameter of 20 meters

step 1: The circumference formula is 2 x 3.14 x r

step 2: divide the diameter by 2 to get the radius: 20/2 = 10

step 3: multiply 2 x 3.14 x 10

*HOW TO FIND THE AREA OF A SQUARE OR RECTANGLE

Area formula for a rectangle: length x width or  a = lw

Area formula for a square: sideor a = s2

example: find the area of the rectangle with the length of 8 and a width of 6.

l

Step 1: Multiply the length and the width : 8 x 6

step 2: the area is 48 units2

example: find the area of the square with the side length of 8.

step 1: Multiply the side length of 8 by the side length of 8: 8 x 8

step 2: the area is 64 units2

*HOW TO FIND THE AREA OF A PARALLELOGRAM

Example: Find the area of a parallelogram with a base of 15 inches and a height of 8 inches

Step 1: Use the formula a = bh

Step 2: replace the b with the number 15 and the h with the number 8

Step 3: mutliply base and height: 15 x 8

Step 4: the area is 120 inches2

*HOW TO FIND THE AREA OF A TRAPEZOID

Example: Find the area of the trapezoid with bases of 6 and 10 and a height of 5

Step 1: use the formula ½ h( b1 + b2)

Step 2: Add the bases together (6+ 10) = 16

Step 3: Multiply the sum of the bases and the height 16 x 5 = 80

Step 4: Divide the product by 2 ( 80/2 = 40)

Step 5: the area is 40 units2

*HOW TO FIND THE AREA OF A TRIANGLE

Example: Find the area of the triangle with a base of 8mm and the height of 9mm.

Step 1: Use the formula ½ bh

Step 2: replace the b with 8 mm and the h with 9 mm

Step 3: multiply 8 and 9 together and then divide the product by 2.

Step 4: the area of the triangle is 36 mm2

*HOW TO FIND THE  VOLUME AND SURFACE AREA OF A RECTANGULAR PRISM

Example: Find the surface area of a rectangular prism with a length of 11cm, a width of 5 cm, and a height of 6cm.

Step 1: Use the formula sa = 2lw + 2lh + wh

Step 2: replace the l with 11, replace the w with 5, and replace the h with 6

Step 3: 2 x 11 x 5 + 2 x 11 x 6 + 2 x 5 x 6

Step 4: The surface area is 302 cm2

Example: Find the volume of a rectangular prism with a length of 11cm, a width of 5 cm, and a height of 6cm.

Step 1: use the formula v = lwh

Step 2: replace the l with 11, replace the w with 5, and replace the h with 6

Step 3: 11 x 5 x 6

Step 4: The volume is 330 cm3

*HOW TO READ INEQUALITIES

x < 3 : this means that x can be any number that is less than 3. For example 2 can be a solution 0 can be a solution -10 can be a solution.

> 20 : this means that x can be any number that is greater than 20 and 20 can also be a solution.

*HOW TO GRAPH INEQUALITIES

STEP 1: Draw a circle around the solution

STEP 2: If it is a greater than or equal to sign or less than or equal to sign shade the circle in. If it is a less than sign or greater than sign leave the circle open.

STEP 3: Draw an arrow in the direction of the solution set. (if the solution set is less than the variable draw the area to the left of the circle. If the solution set is greater than the variable draw the arrow to the right of the circle.

example of    x< 2       x < -4      x > -3

*HOW TO EVALUATE INEQUALITIES

VOCABULARY:

Greater than  >      Less than  <        Greater than or equal to >       Less than  or equal to <

STEP 1: Use inverse operations to isolate the variable.

Example 1: x + 4 < 15 ( the inverse of adding 4 is to subtract 4)

x + 4 < 15       ⇒   x  < 15 – 4       ⇒       x  < 11

Example 2:  a ÷ 10 > 30 (The inverse of dividing by 10 is to multiply by 10)

a ÷ 10 > 30      ⇒  a > 30(10)      ⇒       a > 300

Example 3:  5x < 4  ( The inverse of multiplying by 5 is to divide by 5)

5x < 40         x < 40 ÷ 5       ⇒       < 8

STEP 2: If the inverse operation is to multiply or divide by a negative number you need to switch the direction of the sign.

Example 1:

-3x > 15 ( the inverse of multiplying by -3 is dividing by -3)

x > 15 ÷ -3   ⇒  x > -5   ⇒    x < -5 (since we divided by a negative number we need to switch the sign direction)

Example 2:

x + -3 < 15 ( the inverse of adding a -3 would be subtracting a -3)

x < 15 – (-3)     ⇒    x < 18  (Since we didn’t need to divide or multiply by a negative number we do not need to                                                          switch the direction of the sign.)

** The only other time you would need to switch the direction of the sign is when you swap the left side and the right side of the inequalitiy

EXAMPLE:   X – 4 < 10  WHEN YOU SWITCH SIDES THIS INEQUALITIY BECOMES:   10 > X – 4

*HOW TO COMBINE LIKE TERMS

LIKE TERMS: Terms that have the same variable and are raised to the same power. For example 3x and -9x are like terms. 8z and -88z are like terms. 4h and 4h² are NOT LIKE TERMS because the second term is raised to the second power and the first term is not.

COEFFICIENTS: A numerical quanitity that is placed before a variable in an algebraic expression. It signifies multiplication.

STEP 1: Organize your like terms. You can color code your like terms or you can rearrange the expression so the like terms are next to each other.

Example: 5h + 11g + 1h - 8g ⇒ 5h + 1h + 11g – 8gâ€‹

STEP2: Combine the coefficents of the like terms. With negative numbers you might want to change subtraction into addition. Wtih fractions you might want to make common denominators. If you have fractions and decimals in the same expression you might want to change all the numbers into fractions or change all the numbers into decimals.

Example: (5 + 1)h + (11 - 8)g ⇒ 6h + 3g.

*HOW TO DISTRIBUTE

STEP 1: Multiply each term by the numbers and/or variables outside of the parentheses.

Example: 2(4x - 3y) = 2(4x) - 2(3y) = 8x - 6y

Example: -(3x + 6) = -(3x) + -(6) = -3x + -6

Example: -3(2x + 6u) = -3(2x) + -3(6u) = -6x  + -18u

*HOW TO SOLVE EQUATIONS

Inverse operations: Addition and Subtraction are inverse operations. Multiplication and Division are inverse operations.

STEP 1: Use inverse operations to get the variable on one side alone.

Example: x – 5 = 34 ⇒ x = 34 + 5 ⇒ x = 39

Example: 4x = 20  ⇒ x = 20/4 ⇒x = 5

STEP 2: Replace your solution with the variable to be sure your answer is correct.

Example: 39 – 5 = 34

Example: 4(5) = 20

*HOW TO FIND THE PERCENT OF A NUMBER

STEP 1: Locate the decimal point in the percent. If there is no decimal point add one to the end of the number

Ex: 34% ⇒ 34.% or 345% ⇒ 345.%

STEP 2: Change the percent into a decimal by moving the decimal point two places to the left. If necessary use zeros as placeholders.

EX: 34.% ⇒.34 or 345.% ⇒ 3.45

STEP 3: Multiply the decimal by the number

*EXAMPLE OF HOW TO FIND THE PERCENT OF A NUMBER

What is the 5.9% of 45?

5.9% ⇒ .059 x 45 = 2.655

*HOW TO FIND MARKUPS/ SALES TAX/ TIP

STEP 1: Locate the decimal point in the percent. If there is no decimal point add one to the end of the number

Ex: 34% ⇒ 34.% or 345% ⇒ 345.%

STEP 2: Change the percent into a decimal by moving the decimal point two places to the left. If necessary use zeros as placeholders.

EX: 34.% ⇒.34 or 345.% ⇒ 3.45

STEP 3: Multiply the decimal by the original price

STEP 4: If it is a markup, sales tax, or tip, ADD the original price and the product(answer) from step 3. If it is a markdown or discount, SUBTRACT the original price from the product(answer) from step 3.

*EXAMPLE OF HOW TO FIND THE MARKUP OF  NUMBER

A shoe store uses a 40% markup on the original price of items. Find the cost of a pair of shoes that originally cost \$63.

Step 1: 40% = 40.% (I added a decimal point)

Step 2: 0.40 (I moved the decimal point over two places to the left)

Step 3: 0.40 x 63 = \$25.20 (I multiplied to figure out how much the shoes will be marked up)

Step 4: \$63 + \$25.20 = \$88.20 (I added the original price and the amount of the markup to find the retail price of the shoes or the selling price)

*EXAMPLE OF HOW TO FIND THE MARKDOWN OF  NUMBER

A shoe store is discounting its shoes 33% off for its holiday sale.  Find the cost of a pair of shoes that originally cost \$63.

Step 1: 33% = 33.% (I added a decimal point)

Step 2: 0.33 (I moved the decimal point over two places to the left)

Step 3: 0.33 x 63 = \$20.79 (I multiplied to figure out how much money the shoes will be discounted)

Step 4: \$63 - \$20.79 = 42.21  (I subtracted the original price and the amount of the discount to find the retail price of the shoes or the selling price)