Download 5th and 6th Grade Math Rules Four-step plan to problem solving

5th and 6th Grade Math Rules
1. Four-step plan to problem solving:
- read the problem
- pick out the important facts and decide what operation(s) to perform
- solve the problem
- examine your answer and see if it makes sense.
2. Changing words to numbers: read, write, and compare.
Example: 2,432.241
Two thousand four hundred thirty-two and two hundred forty-one thousandths
3. Standard form – the common way to write a number.
Example: 1,765
4. Expanded form – writing a number as the values of its digits.
Example: 1,765 = 1000 + 700 + 60 + 5
5. Place value – the value of where the digit is in the number.
Place Value Chart
6. A number line is a kind of graph. It is a straight line marked off into units so that you can locate
numbers. A
number line can help you compare numbers and put them order.
Negative Numbers (-)
Positive Numbers (+)
(The line continues left and right forever.)
7. Fact Family – a group of related facts using the same numbers.
Examples: 2 + 3 = 5
4 X 8 = 32
3+2=5
8 X 4 = 32
5–2=3
32 ÷ 4 = 8
5–3=2
32 ÷ 8 = 4
8. Addend - a number to be added.
9. Sum – the answer in an additional problem.
10.
Minuend – the first number in a subtraction problem.
11.
Subtrahend – the second number in a subtraction problem.
12.
Difference–answer in a subtraction problem.
13. Comparing whole numbers: stack the numbers lining the digits up on the right side and then
compare like place values.
Example: 248 207 217
248
207
217
14.
Adding and subtracting decimals – line up the decimals and add zeros if needed.
Example:
13.483
33.900
+ 29.392
- 10.098
15.
Comparing decimals – stack the numbers, line up the decimals, stick in zeros if needed zeros
if needed and compare like place values.
Example: 3.47 925.2 35.497
3.470
925.200
35.497
16. Factor – a number that is multiplied.
product
17.
Product – the answer in a multiplication problem.
18.
Multiply Decimals – stack lining the numbers on the right side. Do not line up the decimal
points. Multiply to get the answer. Count the decimal places in the problem, start at the right in the
answer, and move to the left the same number of places.
Example:
2.34
X .5
1.170
19.
The exponent of a number says how many times to use the number in a multiplication.
Exponents are also called Powers.
In this example: 82 = 8 × 8 = 64
In the example 53 = 5 × 5 × 5 = 125
20.
Dividend – the amount being divided up
21.
Divisor – the number you divide by
22.
Quotient – the answer in a division problem.
23.
Remainder – the number left after the quotient is found.
Divisor
24.
-
34
Four Division Steps Step 1Step 2Step 3Step 4-
2,466
7 73,848
Divide
Multiply
Subtract
Bring Down
Quotient 4
Dividend
Remainder
David’s
Mouse
Saw
Bread
25. Divisible – when one number can be divided by another and the result is an exact whole
number. Example: 15 is divisible by 3, because 15 ÷ 3 = 5 exactly.
But 9 is not divisible by 2 because 9 ÷ 2 is 4 with 1 left over.
26.
2
Divisibility Rules- a number is divisible by:
if the last digit is an even number (0, 2, 4, 6, 8)
3
if the sum of the digits is divided by 3
4
if the last 2 digits are divisible by 4
5
if the last digit is 0 or 5
6
if the number is divisible by both 2 and 3
9
if the sum of the digits is divisible by 9
128 is
1 29 is not
381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, a nd 10÷3 = 3 1/3) No
1312 is (12÷4=3)
7019 is not
175 is
809 is not
114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes
308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No
1629 (1+6+2+9=18, and again, 18÷ 9=2) Yes
2013 (2+0+1+3=6) No
220 is
221 is not
27.
Dividing a Decimal by a Whole Number – bring the decimal straight up to the quotient.
Zeros may need to be added to the dividend.
10
if the number ends in 0
28.
When dividing a decimal by a decimal – move the decimal in the divisor to the end, move
the decimal in the dividend the same number of places, bring the decimal straight up in the quotient.
Zeros may need to be added to the dividend.
29. Repeating decimal – a decimal with a pattern of repeating digits that continues indefinitely.
The repeating decimal can be written using bar notation.
Example:
30. Equation – a number sentence that contains an equal sign.
31.
Algebraic Expression – a number sentence that contains one or more variables.
32. Variable – a symbol for a number we don't know yet. It is typically a letter like x or y.
Variables are used in formulas, equations, and other algebraic expressions.
Example: in x + 2 = 6, x is the variable.
33. . Order of Operations (Please Excuse My Dear Aunt Sally)
Step 1: Do all operations that are grouped together in parentheses ( ).
Step 2: Simplify exponents. (For example, write 33 as 27.)
Step 3: Work all multiplication or division in order from left to right
Step 4: Work all addition or subtraction in order from left to right
Example :
Solution:
Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.
Step 1:
3 + 6 x (5 + 4) ÷ 3 - 7 = 3 + 6 x 9 ÷ 3 - 7 Parentheses
Step 2:
3+6x9÷3-7
= 3 + 54 ÷ 3 - 7
Multiplication
Step 3:
3 + 54 ÷ 3 - 7
= 3 + 18 - 7
Division
Step 4:
3 + 18 - 7
= 21 - 7
Addition
Step 5:
21 - 7
= 14
Subtraction
34.
Rounding numbers- underline the number that is being rounded, look to the right; if the digit
to the right is 5 or higher the number goes up, if the digit to the right is 4 or less the number stays the
same. Everything to the right becomes zeros, and everything to the left copy down.
Examples: 1,839 = 1,840
328.38 = 328.40
156.498 = 156.50
35.
An estimate is an answer that is not exact. It is close to an exact answer.
Estimate a whole number - go to the furthest digit to the left and round off
36.
Compatible numbers – changing numbers to other numbers that form a basic fact to
estimate an answer. Do not round 1-digit numbers.
Examples: 23 X 4 becomes 20 X 4 = 80
568 ÷ 72 becomes 560 ÷ 70 = 80
37. Front-end Estimation with Addition/Subtraction – add/subtract the digits of the highest
place values while the rest of the digits turn into zeros.
Examples:
194
100
314
300
545
500
+216
+200
-275
-200
+180
+100
300
100
600
38. Estimate fractions, round each fraction to 0, ½, or 1. Use a number line for help.
39.
Commutative Property – you can swap numbers over and still get the same answer when you
add or multiply.
Examples: a + b = b + a
3+6=6+3
a×b=b×a
2×4=4×2
40. Associative Property – it doesn’t matter how you group the numbers (in which you calculate
first) when you add or multiply.
Examples: (a + b) + c = a + (b + c) (2 + 4) + 5 = 2 + (4 + 5)
(a × b) × c = a × (b × c) (3 × 4) × 5 = 3 × (4 × 5)
41. Distribution Property – you get the same answer when you multiply a group of numbers by
something as when you do each multiplication separately, like this:
Examples: (a + b) × c = a × c + b × c
(2 + 4) × 5 = 2×5 + 4×5
(6 - 4) × 3 = 6×3 - 4×3
42. Identity Property of Zero in Addition – a number added to zero will be equal to that same
number.
Examples: b + 0 = b
0+2=2
43. Identity Property of One in Multiplication – a number multiplied by one will be equal to
that same number.
Examples:
nX1=n 1X4=4
44. Zero Property of Multiplication – a number multiplied by zero will be equal to zero.
Examples:
a X 0 = 0 49 X 0 = 0
45.
Range of a data group - Subtract the smallest number from the largest number.
46. Mode of a data group - It is the number or numbers that occur most often. If every number is
used just once, there is no mode.
47. Median of a data group - Stack the numbers in order from least to greatest, knock off pairs
from top to bottom, the number in the middle is the median. If there are two numbers left in the
middle, find the mean (average) of the two.
48. Mean (average) of a data group - Add the numbers and divide by the amount of numbers you
have.
Example: Data 98 88 72 95 88
72
88
Median 88
95
98
49.
Range 98 - 72 = 26
72
88 441 ÷ 5 = 88.2
Mode 88
88
Mean
95
+ 98
441
Prime Numbers – a number that has only two factors: 1 and itself.
2 3 5 7 11 13
17
19
23
29
31
37
41
43
47
53
59
61
67
71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 Etc.
50.
Composite Numbers – a number that has more than two factors.
4
33
57
82
6
34
58
84
8
35
60
85
9
36
62
86
10
38
63
87
12
39
64
88
14
40
65
90
15
42
66
91
16
44
68
92
18
45
69
93
20
46
70
94
21
48
72
95
24
49
74
96
25
50
75
98
26
51
76
99
27
52
77
100
51. Every composite number can be expressed as a product of prime numbers – prime
factorization. To find the prime factorization of a number, use a factor tree.
Example: The prime factorization of 84 is:
84
^
The number 84 is composite.
2 X 42
^
2 X 21
84 = 2 X 2 X 3 X 7 The factors 2, 3, and 7 are prime.
^
3X7
52. Greatest Common Factor (GCF) – the highest number that divides exactly into two or more
numbers. If you find all the factors of two or more numbers, and you find some factors are the same
("common"), then the largest of those common factors is the Greatest Common Factor.
Method : List all the factors of each number. Identify the common factors. The greatest of the
common factors is the GCF.
Example: the GCF of 12 and 30 is 6, because 1, 2, 3 and 6 are factors of both 12 and
30, and 6 is the greatest.
28
54
78
102
30
55
80
104
32
56
81
Etc.
53. Multiple of a Number – what you get when you multiply it by other numbers (such as if
you multiply it by 2,3,4,5, etc). Just like the multiplication table.
Examples: multiples of 3 are 6, 9, 12, 15, 18, 21, etc ...
multiples of 12 are 24, 36, lists 48, 60, 72, etc …
54. Least Common Multiple (LCM) – the smallest of the common multiples. If you have two
(or more) numbers, and you check through their multiples and find the same value in both then that is
a common multiple of those numbers.
Method : List several multiples of each number. Example: LCM of 3 and 5.
The multiples of 3 are 6, 9, 15, ..., and the multiples of 5 are 10, 15, 20, ..., like this:
As you can see on this number line, the first time the multiples match up is 15. Answer: 15
55. Fraction – part of a whole. A number written with the bottom part (the Denominator) telling
you how many parts the whole is divided into, and the top part (the Numerator) telling you how
many you have.
Fractions can have three different types:
The numerator is less than the denominator
Proper Fractions:
Examples: 1/3, 3/4, 2/7
Improper Fractions:
The numerator is greater than (or equal to) the
denominator
Examples: 4/3, 11/4, 7/7 (It is "top-heavy")
Mixed Fractions:
A whole number and proper fraction together
Examples: 1 1/3, 2 1/4, 16 2/5
56.
Fractions with Like Denominators - when the denominators are the same. Ex. 1/5, 3/5
57.
Compare Like Fractions - compare the numerators. Example: 2/7 < 5/7
58.
Fractions with Unlike Denominators – when the denominators are not the same.
Example: 1/3 < 2/5
59. Compare unlike fractions - find the least common denominator (LCD) then compare the
numerators, or cross multiply or change the fractions to decimals. Step --Cross Multiply - To
rewrite an equation like a/b = c/d to the form ad = bc.
60. Least Common Denominator (LCD) – is the least common multiple of two fractions. The
smallest number that can be used for all denominators (the bottom number) of two fractions.
Example: the Least Common Denominator of 1/2 and 2/3 is 6, because they can be written as 3/6
and 4/6.
Finding the Least Common Denominator (LCD)  Find the Least Common Multiple (LCM) of the denominators (this is called the
Least Common Denominator).
the same as the least common denominator
 Change each fraction (using equivalent fractions) to make their denominators
the same as the least common denominator.
 Then solve the fractions (add, subtract, etc)!
Example: What is 1/6 + 7/1 5? The LCM is 30.
61.
Reduce a fraction to lowest terms - find the greatest common factor and divide.
62. Equivalent Fractions - are equal. Some fractions may look different, but are really the same,
for example:
4
2
1
/8
=
/4
=
/2
(Four-Eighths)
Two-Quarters)
(One-Half)
63.
Converting improper fractions to mixed Fractions:




Divide the numerator by the denominator.
Write down the whole number answer
Then write down any remainder above the denominator.
Example: Convert 11/4 to a mixed fraction.
Divide: 11 ÷ 4 = 2 with a remainder of 3
Write down the 2 and then write down the remainder (3) above the denominator (4), like
this: 2 ¾
64.
Converting Mixed Numbers to Improper Fractions –

Multiply the whole number part by the fraction's denominator.

Add that to the numerator

Then write the result on top of the denominator.

Example: Convert 3 2/5 to an improper fraction.
Multiply the whole number by the denominator: 3 × 5 = 15
Add the numerator to that: 15 + 2 = 17
Then write that down above the denominator, like this: 17/5
65. Add or subtract fractions with like denominators - add or subtract the
keep the same denominator. Reduce to the lowest terms (simplify) if necessary.
5/8
+
1/8
=
6/8
=
¾
numerators, but
66. Add or subtract fractions with unlike denominators – rename the fractions using the least
common denominator (LCD), then add or subtract. Reduce to the lowest terms (simplify) if
necessary.
67. Add or subtract mixed fractions with like denominators- add or subtract the fractions, add
or subtract the whole numbers, rename and simplify if necessary.
68. Add or subtract mixed fractions with unlike denominators – find the least common
denominator (LCD), add or subtract the fractions, add or subtract the whole, rename and simplify if
necessary.
69.
Change a fraction to a decimal - Divide the numerator by the denominator.
70.
Change a fraction to a percent - Divide the numerator by the denominator.
Move the decimal two places to the right.
71.
Comparing or ordering a combination of fractions, decimals, and/or percents
- convert all to one form in order to make comparisons.
72. Multiplying two proper fractions - cross check to see if you can reduce, multiply across the
top (numerators), multiply across the bottom (denominators), and rename if necessary.
73. Multiply a proper fraction by a mixed fraction - change the mixed number to an improper
fraction, cross check to see if you can reduce, multiply across the top (numerators), multiply across
the bottom (denominators), and reduce if necessary.
74. Multiply a whole number by a mixed fraction - put the whole number over 1, change the
mixed number into an improper fraction, cross check to see if you can reduce, multiply across the top
(numerators), multiply across the bottom (denominators), and rename if necessary.
75. Multiply a mixed fraction by a mixed fraction - change both fractions to improper fractions,
cross check to see if you can reduce, multiply across the top (numerators), multiply across the bottom
(denominators), and rename if necessary.
76. Reciprocals – To get the reciprocal of a fraction, just turn it upside down. In
swap over the Numerator and Denominator.
Fraction
Reciprocal
3
8
/8
/3
5
6
/6
/5
1
3
/3
/1 = 3
19
7
/7
/19
4
other words
1/4
To find the reciprocal of a Mixed Fraction, you must first convert it to an Improper Fraction, then
switch the numerator and denominator of the improper fraction.
77. Divide Fractions - 3 Simple Steps to Divide Fractions Example 1/2 ÷ 1/4
Turn the second fraction (the one you want to divide by) upside-down (this is now the reciprocal 4/1.
Multiply the first fraction by the reciprocal of the second. 1/2 X 4/1 = 4/2. Simplify the fraction 4/2
=2
78. Divide mixed numbers – change the mixed numbers to improper fractions, flip the second
fraction (reciprocal), cross check to see if you can reduce, multiply across the top (numerators),
multiply across the bottom (denominators), and simplify if necessary.
79.
Percentage (%) – parts per 100. It is a ration that compares a number to 100.
So 50% means 50 per 100
(50% of this box is green)
And 25% means 25 per 100
(25% of this box is green)
80.
Decimals, Fractions and Percentages are just different ways of showing the same value.
Percent
Decimal
Fraction
1
1%
0.01
/100
1
5%
0.05
/20
1
10%
0.1
/10
1
12½%
0.125
/8
1
20%
0.2
/5
1
25%
0.25
/4
1
1
33 /3%
0.333...
/3
1
50%
0.5
/2
3
75%
0.75
/4
4
80%
0.8
/5
9
90%
0.9
/10
99
99%
0.99
/100
100%
1
5
125%
1.25
/4
3
150%
1.5
/2
200%
2
81.
Change a percent to a decimal - move the decimal point 2 places to the left and remove the "%" sign.
From Percent
82.
To Decimal
Change a decimal to a percent - Move the decimal two places to the right and add the "%" sign.
From Decimal
To Percent
83. Change a percent into a fraction – write the percent as a fraction with a denomintor of
Simplify if necessary.
Example 25% = 25/100 = ¼
84. Percent of a number - Change the percent to a fraction or a decimal. Multiply it by the number.
Example:
20% of 2,500
20 % of 2,500
20% = 20/100 or 1/5
20% = 20/100 or 0.2
1/5 of 2,500 = 1/5 X 2,500
0.2 of 2,500 = 0.2 X 2,500
= 500
= 500
85. Estimate with percents - round the percent to a fraction that is easy to multiply.
86.
Bar graph – a graph drew using horizontal or vertical bars to show how large each value is.
87.
Double-bar graph - a bar graph that compares frequency of two sets of data.
88. Histogram - a special kind of bar graph that displays the frequency of data that has been
organized into intervals. The intervals cover all possible values of data.
89. Line Graph –a graph that uses points connected by lines to show how something changes in
value (increases and/ decreases) as time goes by, or something else happens.
John’s Weight Trail
90.
Double Line Graph –shows change (increases and/or decreases)over time for two items of data
91. Circle Graph / Pie Graph – a circular chart divided into sectors, each sector shows the
relative size of each value. Each part is expresses as a percentage of the whole. All percents must
add up 100%. Example:
Favorite Colors of the Class
92. Pictograph - uses symbols (pictures) to compare data. A picture graph has a key that tells the
quantity (amount) that is represented by each picture on the graph.
93. Frequency table - used for organizing a set of data, showing the number of times each item or
number appears.
Number of Cars Registered in Each Household
Number of cars (x)
Tally
Frequency (f)
0
1
2
4
6
5
94.
Stem-and-leaf-plot – the arrangement of data with numbers separated into tens and ones.
The tens digits are the stems. The ones digits are the leaves. They are arranged in each row from
least to greatest. A key is included. Example:
Key: 1/5 = 15
95.
Line Plot - shows data using X’s above a number line. Example:
What is Your Favorite Pet?
X
X
X
X
X
X
X
X
X
X
X
Dog
Cat
Fish
Snake
96. Venn Diagram – organizes data using overlapping circles. Venn diagrams are used to show
relationships among things. When parts of the circles overlap, those parts share a certain
characteristic. The parts that don’t overlap don’t share that characteristic. Example:
97.
When making charts or graphs:
- Scale must include all of the data.
Examples: 0 – 10 0-100 500 – 1,000
- Intervals must be equal and not overlapping.
Examples: groups of 2’s, 5’s, 10’s
- Charts or graphs must have title and labels.
- A break in vertical axis is displayed if needed.
98.
Metric Prefixes and Their Meanings:
Kilo – thousands – 1,000
Hecto – hundreds - 100
Deca – tens – 10
Ones – Base Unit - 1
Deci – tenths – 0.1
Centi – hundredths – 0.01
Milli – thousandths – 0.001
Kayla
Kilo-
Helps
Hecto-
Dirty
Deka-
Brown
Base Unit
Dogs
Deci
Count
Centi-
Money
Milli-
99. Customary length – inches (in), foot (ft), yard (yd), mile (mi)
1 inch
Example: about the diameter of a quarter
1 foot = 12 inches
Example: the length of a ruler
1 yard = 3 feet = 36 inches
Example: the length of a yardstick
1 mile = 1,760 yards = 5,280 feet
Example: ten city blocks
*The symbol for inches is ″ and the symbol for feet is ′.
100. Metric Length – millimeter (mm), centimeter (cm), decimeter (dm), meter (m) decameter
(deca), hectometer (hm), and kilometer (km) *Use step method!!
Examples:
millimeter - thickness of a dime
centimeter - width of your fingernail
meter - height of a kitchen counter, length of a guitar
kilometer - nine times a football field including the end zones
101. Capacity (Volume) - how much a container can hold.
102. Customary Capacity (Volume) – teaspoon (tsp), tablespoon (tbsp), ounces (oz), cup (c), pint
(pt), quart (qt), gallon (gal)
1 teaspoon = 1/3 tablespoon
1 tablespoon = 3 teaspoons = ½ ounce
1 cup = 5 1/3 tablespoons = 16 teaspoons = 8 ounces
1 pint = 2 cups = 16 ounces
1 quart = 2 pints = 4 cups = 32 ounces
1 gallon = 4 quarts = 8 pints = 16 cups = 128 ounces
103. Metric Capacity (Volume) – milliliter (ml), centiliter (cl), deciliter (dl), liter (l), decaliter
(dal), hectoliter (hl), and kiloliter (kilo) *Use step method!!
Examples:
Milliliter – 1/2 of what an eyedropper holds
Liter – a large plastic soda or water bottle
Kiloliter – about the size of a large wading pool
104. Customary Mass (Weight) – ounce (oz), pound (lb), ton (T)
Examples:
1 ounce
Example – gerbil, slice of bread, 5 grapes
1 pound = 16 ounces
Example - guinea pig, hammer
1 Ton = 2,000 pounds = 32,000 ounces
Example – milk cow, small car
105. Metric Mass (Weight) – milligram (mg), centigram (cg), decigram (dg), gram (g), dekagram
(dkg), hectogram (hg), and kilogram (kg) *Use step method!!
Examples:
Milligram – eyelash, wing of a housefly, aspirin
Gram - grasshopper, paper clip, dollar bill, raisin
Kilogram - wild rabbit, high top basketball shoe, teacher’s tape dispenser, encyclopedia, textbook, a
little more than 2 pounds
106. Measures of Time – seconds (s), minutes (min), hour (h), day (d), week (wk), year (yr),
century (c), bicentennial, and millennium
Examples:
1 minute = 60 seconds
1 year = 12 months
1 hour = 60 minutes
1 year = 52 weeks
1 day = 24 hours
1 century = 100 years
1 week = 7 days
1 bicentennial = 200 years
1 month = approximately 4 weeks
1 millennium = 1,000 years
1 year = 365 days
107. Polygons – a plane shape with straight sides and is “closed” (all the lines connect up).
Polygons have two dimensions: length and width.
Polygon
(straight sides)
Not a Polygon
(has a curve)
Not a Polygon
(open, not closed)
If all angles are equal and all sides are equal, then it is a regular polygon, otherwise it is irregular
polygon.
Regular
Names of Polygons
Irregular
Sides
Shape
Triangle (or Trigon)
3
Rectangle
4
Quadrilateral (or Tetragon)
4
Pentagon
5
Hexagon
6
Heptagon (or Septagon)
7
Octagon
8
108. Quadrilaterals – are polygons that have 4 sides and 4 vertices. The sum of the angles of
any quadrilateral is 360 degrees.
A parallelogram has two parallel pairs of opposite sides.
A rectangle has two pairs of opposite sides parallel, and four
right angles. It is also a parallelogram, since it has two pairs of
parallel sides.
A square has two pairs of parallel sides, four right angles, and
all four sides are equal. It is also a rectangle and a
parallelogram.
A rhombus is defined as a parallelogram with four equal sides.
Is a rhombus always a rectangle? No, because a rhombus does
not have to have 4 right angles.
Trapezoids only have one pair of parallel sides. It's a type of
quadrilateral that is not a parallelogram. (British name:
Trapezium)
Kites have two pairs of adjacent sides that are equal.
109. Three-Dimensional Figures (3D) - solid figures that have three dimensions: length, width,
and height.
110. Face – flat surface of a three-dimensional figure
111.
Edge – the line were two surfaces meet.
112. Vertex (vertices) – a point where two or more straight lines meet. Corner
113. Prism – three-dimensional solid object that has two identical ends and all flat sides. The
shape of the ends gives the prism their name. It is a polyhedron which means all sides shall be flat.
Triangular Prisms: have 5 faces, 9 edges, and 6
Crossvertices.
Section:
Rectangular Prisms: have 6 faces, 12 edges, and 8
vertices
CrossSection:
Cubes: have 6 square faces, 12 edges, and 8 vertices.
CrossSection:
Pentagonal Prisms: have 7 faces, 15 edges, and 10
vertices.
CrossSection:
114. Pyramids – three-dimensional solid object where the base a polygon and the sides are triangles
which meet at the top (the apex). They are named after the shape of their base.
115. Base - the lowest part. The surface that a solid object stands on or the bottom line of a shape
such as a triangle or rectangle.
Pyramid
Triangular
Pyramid:
Base
A triangular pyramid has 4 faces, 6 edges, and 4 vertices.
Square
Pyramid:
Pentagonal
Pyramid:
A square pyramid has 5 faces, 8 edges, and 5 vertices.
A pentagonal pyramid has 6 faces, 10, edges, and 6 vertices.
116. Cylinders – a three-dimensional solid object that has two identical flat circular ends and one
curved side. It has the same cross-section from one end to another.
117. Cones - a three-dimensional solid object that has a circular bas and one vertex.
118. Sphere – a three-dimensional solid object shaped like a ball. Every point on the surface is the
same distance from the center. It has no faces, bases, edges, or vertices. It is perfectly symmetrical.
119. Perimeter – distance around a two-dimensional shape. Add all sides to find the perimeter.
Example: the perimeter of this rectangle is a + b + a + b = 2(a + b)
120. Area – the amount of space inside the boundary of a flat two-dimensional object.
measures in square units.
Area is
Area of Plane Shapes
Triangle
Area = ½b×h
b = base
h = vertical height
Square
Area = a2
a = length of side
Rectangle
Area = l×h
l = length
h = height
Parallelogram
Area = b×h
b = base
h = height
Trapezoid
Area = ½(a+b)h
h = vertical height
Circle
Area = πr2
Circumference=2πr
r = radius
121. Areas of a Compound Figure – count the whole boxes, count the partial boxes and divide by
2, add the two totals together. *Use attached sheet for example.
122. Circumference – the distance around the edge of a circle (or curvy shape). It is a kind of
perimeter. Circumference=2πr
123. Pi (π) – The ratio of the circumference of the diameter of a circle. It is approximately equal
to: 3.14… when dealing with whole numbers and 22/7 when dealing fractions.
124. Volume - the amount of three-dimensional space an object occupies. Volume is always
labeled cubic or to the third power.
125. Volume of a Rectangular Prism – Volume = Length × Width × Height
126.
Symb
ol
Symbols Used in Geometry:
Meaning
Definition
Example
Triangle
ABC has 3 equal
sides
A triangle with three equal sides is called an equilateral triangle
Angle
ABC is 45°
The amount of turn formed by two rays with the same endpoint
(vertex).
Perpendicular
AB CD
Two lines that intersect at 90° forming right angles.
Parallel
EF GH
Lines in the same plane that run side by side, but never intersect.
Degrees
360° makes a full
circle
A measure for angles. (Degrees are also used for measuring
temperatures).
Right Angle(90°)
is 90°
Line Segment "AB"
AB
A right angle is and angle which is equal to 90°, one quarter of a
full revolution.
A line with definite endpoints.
Line "AB"
Extends in both directions without end ( infinite).
Ray "AB"
A line with a starting point and one end that goes on indefinitely.
Congruent(same shape and
size)
ABC
DEF
Triangle ABC is congruent to triangle DEF
Similar(same shape, different
size)
DEF
MNO
Triangle DEF is similar to triangle MNO
Intersecting Lines
Lines that meet or cross at a common point.
127. Plane – a flat surface with no thickness that extends forever.
128. Angle - the amount of between each arm.
129. Point – a single location or position.
130. Vertex – the corner point of an angle.
131. Protrctor – an instrument used in measuring and drawing angles.
132.
Type of Angle
Description
Acute Angles
an angle that is less than 90°
Right Angle
an angle that is 90° exactly
Obtuse Angle
an angle that is greater than 90° but less than 180°
Straight Angle
an angle that is 180° exactly
133. Congruent Angles have the same angle in degrees. These angles are congruent. They don’t
have to point in the same direction or be on similar sized lines.
134. Triangles and Their Sides / Angles – There are three special names given to triangles that tell
how many sides (or angles) are equal. There can be 3, 2 or no equal sides/angles.
Equilateral
Triangle
Three equal
sides
Three equal
angles, always
60°
Isosceles
Triangle
Two equal sides
Two equal angles
Scalene
Triangle
No equal sides
No equal angles
135. Triangles and Their Inside Angles – There are three special names given to triangles that
tell you what type of angle is inside:
Acute Triangle
All angles are less than 90°
Right Triangle
Has a right angle (90°). The
longest side is called
hypotenuse.
Obtuse Triangle
Has an angle more than 90°
136. Degrees in any circle - 360°. A circle is named by its center point.
137. Circles and Their Special Names:
Lines
A line that goes from one point to another on the
circle's circumference is called a Chord.
If that line passes through the center it is called a
Diameter.
If a line "just touches" the circle as it passes it is
called a Tangent.
And a part of the circumference is called an Arc.
138. Congruent – having the same shape and size. Two shapes are congruent if you can Turn, Flip
and/or Slide one so it fits exactly on the other.
139. Similar –shapes having the same shape and angles, but the only difference is size.
140. Symmetry - when one shape becomes exactly like another if you flip, slide or turn it. If a
figure can be folded so that its two halves match exactly; it has a line of symmetry. Figures can have
more than one line of symmetry.
141. Coordinate System/Coordinate Grid – a plane in which a horizontal number line and vertical
number line intersect at their zero points.
The point (12,5) is 12 units along, and 5 units
up.
X and Y Axis
The X Axis runs horizontally through zero
The Y Axis runs vertically through zero
The point (0,0) is given a special name called “origin”
Coordinates - are always written in a certain order: the horizontal direction first, then the vertical
direction. This is called an "ordered pair". Example: (4,9) means 4 units to the right, and 9 units up
Example: (0,5) means 0 units to the right, and 5 units up.
142. Survey - To gather information by individual samples so as to learn about the whole thing.
Example: you could survey a river's water quality by taking a cupful of water from different locations
at different times. You can also do a survey on people's opinions, by asking randomly chosen people
the same question.
143. Sample – a selection taken from a larger group (the “population”) so that you can examine it to
find out something about the larger group.
144.
Square Numbers - To square a number, just multiply it by itself. Examples:
3 Squared = 32 = 3 × 3 = 9
4 Squared = 42 = 4 × 4 = 16
5 Squared = 52 = 5 × 5 = 25
145. Square roots – The square root of a number is that special value that, when multiplied by
itself, gives the number. The symbol for square root is √
3 squared is 9, so the square root of 9 is 3.
2
4
3
9
4
16
5
25 etc.
146. Perfect Squares - are the squares of the whole numbers:
Perfect
Squares:
1
2
3
4
5
6
7
8
9
10 etc
1
4
9
16
25
36 49
64
81 100 ...