Chapter 3: Mathematics
Decimal number
A number represented by the digits 0 and 9 and may include a decimal point. (4.5, 0.003, 367.0, etc.). All whole numbers can be written with a decimal point to the right of the ones place.
Irrational number
A number that cannot be expressed as the ratio of two integers. Decimals that never end (called non-terminating) and do not have a repeating pattern are irrational.
Difference between a pyramid and a prism
A pyramid is named by the shape of its base. No matter what the shape of the base of a pyramid, its lateral faces will always be triangular. A prism is named by the shape of its base, and it has two congruent bases.
Sphere
A solid shape that is perfectly round like a ball. No faces, edges, or vertices.
Prism
A solid with lateral faces and bases that are congruent polygons
Prime number
A whole number that has exactly two factors, 1 and itself. Examples: 2,3,5,7,11,13, etc.
Formula for area of triangle
A=1/2bh
Surface area of a cylinder
A=2(πr^2)+2(πr)h
Surface area of a cone
A=πrs+πr^2
Area of circle
A=πr²
When reading a decimal number, you use the word ___________ for the decimal place.
AND. Ex.: The decimal 4.5 is read four AND five tenths, since it ends in the tenths place.
Whole numbers
All the positive integers and 0. (0, 1, 2, 3..........)
Pyramid
Always have triangles for their lateral faces.
Straight angle
Angle of 180 degrees (straight line)
Right angle
Angle whose measure is 90 degrees.
Reflex angle
Angle whose measure is greater than 180, less than 360.
Acute angle
Angle whose measure is larger than 0 degrees but less than 90 degrees.
Obtuse angle
Angle whose measure is larger than 90 degrees but less than 180 degrees.
Congruent angles
Angles of equal measure
Prime factorization
Breaking down a composite number until all of the factors are prime
Circumference Formula
C = 2πr
Perfect square numbers
Numbers that result form multiplying an integer by itself. The first. The first 10 square numbers are 1,4,9,16,25, 36, 49, 64, 81, and 100.
Vertices
Point where multiple faces meet
Vertex
Point where two lines meet.
Quadrilateral
Polygon made up of four straight lines.
Natural or counting numbers
Positive integers beginning with 1 (1, 2, 3, 4, .........)
Area of Quadrilaterals
Rectangle: A=lw Parallelogram: A=bh Trapezoid: A=1/2h(b1+b2)
Surface area of a sphere formula
SA=4πr²
Congruent
Same size and shape.
Diameter
Segment from one point on the circumference to another point on the circumference passing through the center of the circle.
Radius
Segment from the center of a circle to a point on its circumference.
Congruent figures are also considered ________________.
Similar.
Cylinder
Solid with bases that are congruent circles
Key words for multiplication
1. Multiply 2. Times 3. Product
PEMDAS
1. Please (Parenthesis) 2. Excuse (Exponent) 3. My (Multiplication) 4. Dear (Division) 5. Aunt (Add) 6. Sally (Subtract)
Key words for subtraction
1. Subtract 2. Minus 3. Difference 4. Less than 5. Decreased by 6. Take away 7. Fewer
Types of Quadrilaterals
1. Trapezoid 2. Parallelogram 3. Rhombus 4. Rectangle 5. Square
Ratio
A comparison of two quantities. Ratios can be written in three different ways: - With the word "to": 4 to 5 - With a colon: 4:5 - As a fraction: 4/5
Properties of a Rhombus
- All four sides congruent
Properties of a Trapezoid
- One pair of parallel sides
Number of sides in polygon: Name of polygon: 3 __________________________ 4 __________________________ 5 __________________________ 6 __________________________ 7 __________________________ 8 __________________________ 9 __________________________ 10 __________________________
- Triangle - Quadrilateral - Pentagon - Hexagon - Septagon - Octagon - Nonagon - Decagon
Properties of a square
- Two pairs of parallel sides - All four sides congruent - Four right angles
Properties of a Parallelogram
- Two pairs of parallel sides - Opposite sides congruent
Properties of a Rectangle
- Two pairs of parallel sides - Opposite sides congruent - Four right angles
Ellipse
A elongated circle, or oval shape, the shape of the planets orbit.
Rules for exponents
1. A^b x A^c = A^b+c 2. A^b/A^c = A^b-c 3. (A^2)^4 = A^2x4 4. 1/A^-b=A^b
Key words for addition
1. Add 2. And 3. Plus 4. More than 5. Increased by 6. Sum 7. Total
Key words for division
1. Divide 2. Quotient
Converting decimals, percents, and fractions.
1. Fraction ------> decimal (if denominator is power of 10) 2/10=0.2. 17/100=0.17. 2. Decimal ------> fraction: write the numbers of the decimal in the numerator of a fraction. Locate the digit that is farthest to the right in the number, write the place value in the denominator. Make sure to simplify. 0.55=55/100=11/20 3. Percent --------> decimal: Percent sign means per hundred, so it's calculated by dividing the original number by 100. Move the decimal point in the number two places to the left. REMEMBER. IF IT'S A WHOLE NUMBER, DECIMAL POINT IS STILL AT THE END. 19%=19.%=0.19. 4. Decimal---------> percent: Move the decimal two places to the right and add a percent sign. 0.64=64%. 0.578=57.8%. 5. Percent-------> fraction: Write the number in front of the percent sign in the numerator, then use 100 as denominator. SIMPLIFY. 14%=14/100=7/50. 6. Fraction ------------> percent: First divide the numerator by the denominator. Then move the decimal point two places to the right. Ex: 1/8=0.125. 0.125=12.5%. 7. Fraction--------> decimal: Just divide the number.
How to write a fraction as a decimal if the denominator is a power of 10.
2/10=0.2 17/100=0.17 45/1000=0.045
What to do if we have a negative exponent (ex. 3^-2)
3^-2= 1/3^2= 1/(3x3)= 1/9
Writing 0.55 as a fraction.
55/100. Then we would simplify it to 11/20.
Cone
A "pyramid" with a circular base is called a cone.
Polygon
A closed plane figure made up of line segments
How to decide whether a fraction is larger or smaller than another.
CROSS MULTIPLY. Multiply the denominator of first fraction with numerator of second fraction. Multiple the numerator of the first fraction with the denominator of the second fraction.
Types of triangles
Categorized by sides: 1. Equilateral: all equal sides 2. Isosceles: Two equal sides 3. Scalene: No equal sides. Categorized by angles: 1. Acute: Three acute angles 2. Obtuse: One obtuse angle 3. Right: One right angle
Angle
Collection of points that is the union of two rays having the same endpoint.
How to compare negative fractions (same denominator)
Compare the numerator. Ex: -2/9 and -7/9. Because -2 is greater than -7, -2/9 is bigger.
Real numbers
Comprised of rational and irrational numbers. (23, -12, 6.99, 5/2, π, etc.)
Rectangular solid
Congruent polygons and lateral faces are all rectangles.
How to compare negative fractions (different denominator)
Cross multiply (assign each fraction's negative sign to its numerator). Ex: -3/4 and -7/8. -24 is greater than -28, so -3/4 is greater than -7/8.
Circumference
Distance around the circle (think of it as the perimeter of the circle)
Divisibility rules
Divisible by 1: All whole numbers are divisible by 1. Divisible by 2 (also called even): If a number ends in 0,2,4,6,8, it is an even number, so it's divisible by 2. Divisible by 3: Add up the digits of the number; if that sum is divisible by 3, then the number is divisible by 3. Example: 81, 8+1=9. 9 is divisible by 3, so 81 is divisible by 3. Divisible by 4: If the last two digits of a number are divisible by four, the number is divisible by 4. Divisible by 5:A number is divisible by 5 if it ends in 5 or 0. Divisible by 6: A number that is divisible by 2 and 3 is also divisible by 6. Divisible by 8: If the last three digits of a number are divisible by 8, the number is divisible by eight. Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Divisible by 10: If a number ends in 0, it's divisible by 10.
Writing improper fractions as a mixed number.
Example: Write 27/4 as a mixed number. 1. Divide the denominator into the numerator: 27 divided by 4=6 remainder 3. 2. Write answer as the whole number: 6 3. Make remainder the new numerator, use old denominator. 6 3/4.
Writing mixed numbers as improper fractions.
Example: Write 5 3/8 as an improper fraction. 1. Multiple the denominator of the fraction by the whole num/8=0.125.ber (8x5=40) 2. Add the answer to the numerator (40+3=43) 3. Write the answer as the new numerator over original denominator (43/8)
Another way to compare fractions WITHOUT CROSS MULTIPLYING
Find the least common multiple for the common denominator. Then compare the numerators, list from least to greatest.
Writing a fraction as a percent.
First divide the numerator by the denominator. Then move the decimal point two places to the right. Ex: 1/8=0.125. 0.125=12.5%.
Faces
Flat surfaces of three-dimensional figures
What to do if we have an exponent of 0.
NUMBER BECOMES 1.
Vertical angle
Formed when two lines intersect. The angles are equal.
Composite numbers
If a number has more than two factors, it's composite.
What to do if we have an exponent of 1.
NUMBER STAYS THE SAME.
Fractions
Numbers that can be used to express parts of a whole. A fraction has a numerator and a denominator. - The denominator is the number on the bottom, and it shows how many pieces the fraction is broken up into. - The numerator shows how many parts of the fraction you have. (1/5, 3/2, 6/7, 3/4, etc.)
Rational numbers
Numbers that can be written as a ratio of two integers. Rational numbers include integers, fractions, and decimals. (1/3, 2/4, 1/5, 9/3, 1.3, 2.7, etc.)
Factors of a number
Numbers that divide into it
Edge
Line segment where two faces meet
Integers
The set of whole numbers and their opposites {. . .-2, -1, 0, 1, 2. . .}.
Changing a decimal to a percent.
The decimal moves two places to the right, and the % sign is added. Ex: 0.50=50%, 0.25=25%.
Greatest common factor
The largest factor that two or more numbers have in common.
least common multiple (LCM)
The least number that is a common multiple of two or more numbers.
Similar triangles
The measures of the angles of a pair of similar triangles are equal in a one-to-one fashion. A triangles whose angles are 30, 60, and 90 is similar to every other triangle with those angle measurements, even though the sides of the two triangles may be different. The sides, however, are proportional, meaning they correspond to one another.
Converting a percent to a decimal.
The percent sign means "per hundred", so we calculate percent by dividing the original number by one hundred. This is done by moving the decimal point in the number two places to the left. IF THERE IS NO DECIMAL POINT, IT MEANS THE NUMBER IN FRONT OF THE PERCENT SIGN IS A WHOLE NUMBER. For whole numbers, the decimal point is directly after the whole number. EX.: 19%=19.% 0.19.
Scale factor
The ratio of the lengths of two corresponding sides of two similar polygons
Square root
To find the square root of a number, you must find a number that when squared, equals your original number.
Adjacent angle
Two angles with a common vertex and common side but no common interior points. DBC and ABC are adjacent angles in the diagram.
Supplementary angles
Two angles, sum of the measures of which equals 180.
Complementary angle
Two angles, the sum of the measures of which equals 90 degrees.
Equivalent fractions
Two fractions that are equal to each other, but have different denominators; equivalent fractions simplify to the same fraction written in lowest terms. Ex: 2/5=4/10. 1/4=25/100. 2/3=6/9.
How many square roots does each number have?
Two. A positive and a negative one. The square root of 81 is 9, but it can also be -9, because -9x-9 is still 81.
Finding percent of a number.
Use part over whole technique. Ex.: What percent of 80 is 5? 5/80=0.0625=6.25%.
Pythagorean theorem
Used to find the missing side length of a right triangle.
Volume of a cone
V=1/3πr^2h
Volume of a sphere formula
V=4/3πr³
Volume of a cylinder
V=Bh
Raised to the power
When a number is multiplied by itself a specific number of times.
Proportion
When two ratios are equal to each other.
Difference between whole and natural numbers?
Whole STARTS with 0!!! Natural STARTS with 1!!!!!! Natural numbers start with 1 because it is NATURAL to start with one when. you are counting items.
Writing a percent as a fraction.
Write the number in front of the percent sign in the numerator of the fraction, then use 100 as the denominator. 14%=14/100=7/50.
Writing a decimal as a fraction.
Write the numbers of the decimal in the numerator of a fraction without the decimal point. Locate the digit that is farthest to the right in the number and write its place value as the denominator. If the fraction needs to be simplified, then rewrite it in lowest terms.
a^n=b. (three different definitions)
a=base n=exponent or power that shows the number is times the base is to be multiplied by itself b=product of the multiplication.
